### Speedy Sidney

Two trains set off at the same time from each end of a single straight railway line. A very fast bee starts off in front of the first train and flies continuously back and forth between the two trains. How far does Sidney fly before he is squashed between the two trains?

### Illusion

A security camera, taking pictures each half a second, films a cyclist going by. In the film, the cyclist appears to go forward while the wheels appear to go backwards. Why?

### Walk and Ride

How far have these students walked by the time the teacher's car reaches them after their bus broke down?

# An Unhappy End

### Why do this problem?

This problem only requires some simple understanding of the relationship between time, distance and speed, but it will require clear thinking and insight to solve. It provides a good experience of the mathematical 'lightbulb' moment when a seemingly tricky problem is solved using a clever trick or alternative viewpoint.

### Possible approach

Show the 'film' asking pupils to focus on the fly. Ask them what data they would need to analyse the situation mathematically, and provide the speeds and distance.

Ask students to begin calcuating times, distances and positions of the trains and the fly. They will need to establish a clear way of recording their results. Encourage them to describe their strategy, even if they are frustrated by the length of the task.

It is useful to discuss with students that in some circumstances, the only option for solving a problem is to find a long, complex algorithm. However, the frustration CAN be a trigger to step back and find more efficient methods, and in this case there is a good method.

The simple way of looking at this problem is the total time until collision of the two trains and the complicated way is working out the points at which the fly hits a train on each part of the journey and then adding these up. Ideally someone in the group will come up with the simpler method - if not, you could leave it as an open problem, or begin to encourage the group to focus on time.

### Key questions

• How long will it be until the fly first hits a train?
• How long will it be until the trains collide?

### Possible extension

Alter the distance between the engines or the speeds to change the problem.
If the fly flew for 1 minute, what could the distance and speeds have been?
If the fly flew 2000 metres, what could the distance and speeds have been?

Speedy Sidney provides a similar context for a slightly more challenging problem.

This problem is a perfect opportunity to think about infinite series - the fly makes infinitely many trips between the two trains before collision. The total time taken for all of the trips must equal the time taken for the trains to collide. Students might wish to investigate the creation of this series by working out how long the fly takes to travel on each portion of its journey.

### Possible support

As a warm up, you might try the simpler problem Bike Ride